Math Problem Statement

Solve for x in terms of a, b, and c. If there is more than one solution, separate your answers with a comma. The equation is: ax^2 + bx + c = 0.

Solution

The image you've uploaded shows a mathematical problem asking to solve for a variable in terms of other variables. Here is a transcription of the problem:

Problem: Solve for xx in terms of aa, bb, and cc. If there is more than one solution, separate your answers with a comma.

The equation given is:

ax2+bx+c=0ax^2 + bx + c = 0

Solution:

The equation provided is a quadratic equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. To solve for xx, we use the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Explanation:

  1. Quadratic Formula: This formula gives the solutions for a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.
  2. Discriminant: The term inside the square root, b24acb^2 - 4ac, is called the discriminant and determines the nature of the roots:
    • If b24ac>0b^2 - 4ac > 0, there are two distinct real roots.
    • If b24ac=0b^2 - 4ac = 0, there is one real root (a repeated root).
    • If b24ac<0b^2 - 4ac < 0, there are two complex roots.

Final Answer:

x=b+b24ac2a,x=bb24ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}, \quad x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

These are the two possible solutions for xx in terms of aa, bb, and cc.

Would you like a more detailed explanation or have any questions?


Related Questions:

  1. How do you determine the discriminant of a quadratic equation and what does it signify?
  2. What are the steps to complete the square for a quadratic equation?
  3. How do you solve a quadratic equation graphically?
  4. How can you find the vertex of a quadratic function using its standard form?
  5. What are the real-life applications of solving quadratic equations?

Tip:

Remember that the discriminant helps determine the type of roots without actually solving the quadratic equation!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Roots of Equations

Formulas

Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Quadratic Formula

Suitable Grade Level

Grades 9-11